The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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2.3. Going to high er energies: a toy V(x, x'); [fm-4] V(x, x') [fm-4] 0 -0.05 - 0 .1 -0.15 -1 -0.2 model 47 X ' [fm] [fm] Figure 2.12: Coordinate space representation of the effective S-wave potential Vo(x, x') . Left panel: A = 400 MeV and 0 ::; x,x' ::; 4fm. Right panel: A = 5.5GeV and 1.2 ::; x,x' ::; 2fm.
Chapter 3 The derivation of nuclear forces from chiral Lagrangians 3.1 Chiral symmetry Let us start with the QCD Lagrangian for Nf quarks q LQCD 1 qif/Jq - 4 G�vGQ' JlV - qMq (3.1) L�CD + L�CD ' where L�CD is the Lagrangian for massless quarks given by the first two terms in the equation (3.1) and L�CD denotes the quark mass term. To simplify the notation we do not show explicitly the color indices of quark fields. Note that the Nf x Nf fiavor matrix M can be brought into a diagonal form by an appropriate unitary transformation on quark fields in the fiavor space. The covariant derivative of the quark field DJlq is defined via D Jlq = (OJl + igA� >' 2 Q) q (3.2) where >' Q are 3 x 3 matrices in the color space. They satisfy the SU(3) Lie algebra Furt her , 9 is the strong coupling constant and AJl is the gluon field. Whereas quarks belong to the fundamental representation of the SU(3)col group (color triplet), gluon fields form the adjoint representation of that gauge group (color octet). The field strength tensor G JlV is defined as GQ _ !Cl AQ !Cl AQ jQ ß 'YA ß A'Y JlV - UJl v - Uv Jl - 9 Jl v ' where jQ ß 'Y are the SU(3) structure constants introduced in eq. (3.3). The QCD Lagrangian for massless quarks L�CD can be rewritten in terms of left- and righthanded quark fields qL == PLq, qR == PL q, where PR,L are the corresponding projectors as follows: 48 (3.3) (3.4) (3.5) (3.6)
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The Nucleon-Nucleon Interaction in
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11 vom ursprünglichen wesentlich u
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IV Ordnung des Potentials besteht a
Page 7 and 8: VI sagen für die Schwerpunktsimpul
Page 9 and 10: Vlll betrachten, um eine komplette
Page 11 and 12: x 5 Isospin violation in the two-nu
Page 13 and 14: 2 1. Introduction nuclear force of
Page 15 and 16: 4 1. Introduction the Thcson-Melbou
Page 17 and 18: 6 1. Introduction pion-nucleon syst
Page 19 and 20: 8 1. Introduction completely domina
Page 21 and 22: 10 1. Introduction such interaction
Page 23 and 24: 12 2. Low-momentum effective theori
Page 39 and 40: 28 2. Low-momentum efIective theori
Page 47 and 48: 36 2 o -2 -4 -6 -8 -10 � -12 t-
Page 49 and 50: 38 (Eq - Ep )A(p, q) [GeV-2] 2 1.6
Page 53 and 54: 42 -3 -5 -7 -9 2. Low-momentum effe
Page 57: 46 2. Low-momentum effective theori
Page 61 and 62: 50 3. The derivation of nuc1ear for
Page 63 and 64: 52 and for the SU(Nf h x SU(Nf)R [Q
Page 65 and 66: 54 3. The derivation of nuclear for
Page 77 and 78: 66 3. The derivation o{ nuclear {or
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Page 95 and 96: 84 .... . - - .... \ . • A .... .
Page 105 and 106: 94 , , , , , , , , , , , , , , , ,
Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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98 3. The derivation o{ nuc1ear {or
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100 3. The derivation o{ nuc1ear {o
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102 3. The derivation o{ nuclear {o
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104 3. The derivation of nuclear fo
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114 / / / / / / / 5 / / \ \ \ \ I I
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116 3. The derivation o{ nuc1ear {o
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120 4. The two-nuc1eon system: nume
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134 10 8 4 2 o o • Nijmegen PSA N
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136 4. The two-nucleon system: nume
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140 4. The two-nucleon system: nume
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146 --- .... . � - --- .... . ...
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148 � .... --- � � .... --- :
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4.5. Results for the NNLO-ß approa
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5.1 Isospin violating effective Lag
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5.1 Isospin violating effective Lag
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5.2 Brief introduction into the KSW
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5.3 The leading eIB and eSB effects
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5.3 The leading CIB and CSB effects
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5.5 Classification scheme 163 scatt
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Chapter 6 Summary and outlook In th
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2. Secondly, we considered the real
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NNLO, this range is larger and exte
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might be responsible for solving th
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derivative. Further helpful equalit
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Appendix B Operators contributing t
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Appendix C Small scale expansion wi
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179 (C.16) l�r-2. (C.17) h + l2 =
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Appendix E Anti-symmetrization of t
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Consequently, only four of the eigh
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where qJ-! is chosen to satisfy (v
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To keep the presentation self-conta
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- i�03{ [(NtVN) . (VNt x ifN) + (
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Consequently, it is not possible to
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(j ± 1, 1jIVIJ =f 1, 1j) Here, Pj
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Bibliography [1] E. Rutherford, Phi
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[62] J. Goldstone, A. Salam and S.
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[137] V. Bernard, N. Kaiser and Ulf
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[204] J.M. Blatt and L.C. Biedenhar
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The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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